
In [1]}, Shalika proved that the local and global generic Whittaker model, whenever exists, has multiplicity 1. In particular, Whittaker model always exists for cuspidal automorphic representation. Based on this, [1]} and [3]} proved the local and global multiplicity-one theorem for \(GL(n)\) . The existence of global Whittaker model is also important in the construction of automorphic \(L\) -functions with the Langlands-Shahidi method.

In [4]}, Ginzburg proved that the maximal orbit allowing a nonzero Fourier coefficient for the generalized Speh representation (as a residue of cuspidal Eisenstein series) \(\Delta (\tau ,n)\)  is \([a^n]\)  by showing the vanishing of the induced local Whittaker functional. In [5]}, the same result was proven following a global argument based on the root exchange technique. A recipe for the root exchange technique is available in Section 7.1 of [6]}, and a systematic reduction method based on the root exchange technique is developed by Gourevitch et al. in [7]}. A similar global method also appears in [8]} for the orbit of the top Fourier coefficient of degenerate Eisenstein series of \(GL(n)\) , and in [9]} for the orbit corresponding to the top Fourier coefficient of isobaric sum representations.

